If $A\subseteq B$ is a module-finite extension of domains, when $[$Frac$(A):{}$Frac$(B)]=$ number of generators of $B$ over $A$

Question

This is a follow up question to another asked earlier here. From the answer there, it follows that if $A\subseteq B$ is a module-finite extension of domains, then $\operatorname{Frac}(B)$ is generated over $\operatorname{Frac}(A)$ by a smaller number of elements than the number required to generate $B$ over $A$. I was wondering when does equality hold.